78.29.3 problem 1 (c)

Internal problem ID [18411]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 60. Critical Points and Stability for Linear Systems. Problems at page 539
Problem number : 1 (c)
Date solved : Monday, March 31, 2025 at 05:27:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 30

ode:=[diff(x(t),t) = -3*x(t)+4*y(t), diff(y(t),t) = -2*x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{t}+\frac {c_2 \,{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 67

ode={D[x[t],t]==-3*x[t]+4*y[t],D[y[t],t]==-2*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} \left (2 c_2 \left (e^{2 t}-1\right )-c_1 \left (e^{2 t}-2\right )\right ) \\ y(t)\to e^{-t} \left (c_2 \left (2 e^{2 t}-1\right )-c_1 \left (e^{2 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.075 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{- t} + C_{2} e^{t}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t}\right ] \]